Complex algebraic variety

The Riemann sphere is one of the simplest complex algebraic varieties.

In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers.

Chow's theorem

Chow's theorem states that a projective analytic variety; i.e., a closed analytic subvariety of the complex projective space [math]\displaystyle{ \mathbb{C}\mathbf{P}^n }[/math] is an algebraic variety; it is usually simply referred to as a projective variety.

Hironaka's theorem

Let X be a complex algebraic variety. Then there is a projective resolution of singularities [math]\displaystyle{ X' \to X }[/math].

Relation with similar concepts

Not every complex analytic variety is algebraic, though.

See also

• Complete variety
• Complex analytic variety

Bibliography

• Abramovich, Dan (2017). "Resolution of singularities of complex algebraic varieties and their families". Proceedings of the International Congress of Mathematicians (ICM 2018). pp. 523–546. doi:10.1142/9789813272880_0066. ISBN 978-981-327-287-3. 
• Hironaka, Heisuke (1964). "Resolution of Singularities of an Algebraic Variety over a Field of Characteristic Zero: I". Annals of Mathematics 79 (1): 109–203. doi:10.2307/1970486. https://doi.org/10.2307/1970486.